Number grids

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Introduction

Number grids

My task is to find an algebraic rule for different sized squares in a set sized number grid.

To do this I will establish my algebraic rule by creating a 10×10 square and marking out 3 different sized squares inside this square. I will then work out the rules for these individual squares and combine them to create my overall rule.

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I have marked out my smaller squares inside the grid and will now work out an algebraic rule:

To find my algebraic rule I will times the opposite corners in the inset squares and take the numbers away from each other to find the difference.

2×2-

55×66=3630

56×65=3640

3640-3630 = 10

89×100=8900

90×99=8910

8910-8900=10

22×33=726

23×32=736

736-726=10

27×38=1026

28×37=1036

1036-1026=10

After multiplying the corners of the 2×2 squares I then took the lowest away from the highest. This number is always 10.

In this section: a represents the number in the top left hand corner of the inset square.

a

a+1

a+10

a+11

(a+1)×(a+10)-a×(a+11)Here I have multiplied the opposite corners of the grid

[a²+11a+10]-[a²+11a] Here I have multiplied out the brackets and simplified the rule

a²+11a+10 -

a²+11a _ Here I have subtracted the two sections to prove my overall rule.

Related GCSE Number Stairs, Grids and Sequences essays

I'll try it for a 2 x 4 cutout. I predict that the diagonal difference will be 30 So in the case of a 2 x 4 cutout the diagonal difference = (4-1)10 = 30 My prediction was correct But this solution will not work for a vertically aligned cutout as there is more than one G.

I think that by using this formula you will be able to find out the difference in any size rectangle in any size square grid. I think that this formula will also work to find out any size square in an any size square grid.

2 ). This rule can be explained better using algebraic formulae for the two products: (n = number in the top left hand corner) Top left/bottom right product: n(n + 11) = n� + 11n Top right/bottom left product: (n + 1)(n + 10)